Sharp Bounds of the Hermitian Toeplitz Determinants for Certain Close-to-Star Functions

Theorem 1.1 [8] Let F be a subclass of A such that { f ∈F : a2 = 0} ̸= ∅and A2(F) := max{|a2| : f ∈F} exists. Then, 1 −A2 2(F) ≤det T2,1( f ) ≤1. Both inequalities are sharp. Let S∗denote the subclass of S of starlike functions, i.e., f ∈S∗if f ∈A and Re zf ′(z) f (z) > 0, z ∈D. A function f ∈A is called close-to-star if there exist g ∈S∗and β ∈R such that Re eiβ f (z) g(z) > 0, z ∈D. (1.2)

Lemma 1.2 If p ∈P is of the form (1.6), then |cn| ≤2, n ∈N. (1.7) Moreover, c1 = 2ζ1 (1.8) and c2 = 2ζ 2 1 + 2(1 −|ζ1|2)ζ2 (1.9) for some ζi ∈D, i ∈{1, 2}. For ζ1 ∈T, there is a unique function p ∈P with c1 as in (1.8), namely, p(z) = 1 + ζ1z 1 −ζ1z ,

Theorem 2.1 If f ∈F1, then −8 ≤det T2,1( f ) ≤1. Both inequalities are sharp. Now, we will compute the bounds of det T3,1( f ) in the class F1.

Theorem 2.2 If f ∈F1, then det T3,1( f ) ≤39. (2.4) The inequality is sharp.

Theorem 2.3 If f ∈F1, then det T3,1( f ) ≥1 8  6 √ 2 −15  4 + 2 √ 2  2 3 +

Theorem 3.1 If f ∈F2, then −8 ≤det T2,1( f ) ≤1. Both inequalities are sharp. Now, we estimate det T3,1( f ) in the class F2.

Theorem 3.2 If f ∈F2, then det T3,1( f ) ≤48. (3.4) The inequality is sharp.

Theorem 2.2, the inequality (2.5) holds with the function F(x, y) := 2x2y −2x2 −y2 + 1, (x, y) ∈[0, 3] × [0, 5] . Repeating argumentation in the in the proof of Theorem 2.2, we see that the function F does not have any relative maxima in (0, 3) × (0, 5). We consider F on the boundary of [0, 3] × [0, 5]. (1) On the side x = 0, F(0, y) = 1 −y2 ≤1, y ∈[0, 5]. (2) On the side x = 3, F(3, y) = −17 + 18y −y2 ≤F(3, 5) = 48, y ∈[0, 5]. (3) On the side y = 0, F(x, 0) = 1 −2x2 ≤1, x ∈[0, 3].

Theorem 3.3 If f ∈F2, then det T3,1( f ) ≥−4 7